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Let G be a p-mixed abelian group with semi-complete torsionsubgroup G_t such that G is splitting or is of torsion-freerank one, and let R be a commutative unitary ring of primecharacteristic p. It is proved that the group algebras RG andRH are R-isomorphic for any group H if and only if G andH are isomorphic. This isomorphism relationship extends ourearlier results in (Southeast Asian Bull. Math., 2002), (Proc.Amer. Math. Soc., 2002) and (Bull. Korean Math. Soc., 2005) aswell as completely settles a problem posed by W. May in (Proc.Amer. Math. Soc., 1979).